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  "body": "\n## Definition\n\nLet $n \\ge 0$ be an integer and let $X$ be a [[Hausdorff|0MXB4CKHFW]] and [[second-countable|00RE1R4W7N]] topological space.\nAn *orbifold chart* of dimension $n$ on an open set $U \\subseteq X$ is a triple $(\\tilde U, G, \\varphi)$ with these parts.\nThe set $\\tilde U$ is a connected open subset of $\\R^n$.\nThe group $G$ is finite and has a [[faithful|0B0WE1GKTK]] [[action|0YSE802YCQ]] on $\\tilde U$ by diffeomorphisms.\nThe map $\\varphi \\colon \\tilde U \\to U$ is continuous, and $\\varphi(g x) = \\varphi(x)$ for all $g \\in G$ and $x \\in \\tilde U$.\nThe induced map from the [[quotient space|0V8046R0P9]] $\\tilde U / G$ to $U$ is a [[homeomorphism|05GZ11PWRY]].\nAn *embedding* of a chart $(\\tilde U_i, G_i, \\varphi_i)$ into a chart $(\\tilde U_j, G_j, \\varphi_j)$ is a smooth embedding $\\psi \\colon \\tilde U_i \\to \\tilde U_j$ with $\\varphi_j \\circ \\psi = \\varphi_i$.\nAn *orbifold atlas* of dimension $n$ on $X$ is a family of charts $(\\tilde U_i, G_i, \\varphi_i)$ with these properties:\n\n1. The sets $U_i = \\varphi_i(\\tilde U_i)$ cover $X$.\n2. For any two charts and any point $x \\in U_i \\cap U_j$, there is a chart $(\\tilde U_k, G_k, \\varphi_k)$ of the atlas with $x \\in U_k \\subseteq U_i \\cap U_j$.\n3. Whenever $U_i \\subseteq U_j$, there is an embedding of the chart $(\\tilde U_i, G_i, \\varphi_i)$ into the chart $(\\tilde U_j, G_j, \\varphi_j)$.\n\nAn atlas $\\mathcal V$ *refines* an atlas $\\mathcal U$ if every chart of $\\mathcal V$ has an embedding into some chart of $\\mathcal U$.\nTwo atlases are equivalent if they have a common refinement.\nAn $n$-dimensional *orbifold* is a space $X$ as above together with an equivalence class of $n$-dimensional orbifold atlases.\n\n## Remarks\n\nAn embedding $\\psi$ of charts induces an injective [[group homomorphism|0HV9Y7WG5R]] $f \\colon G_i \\to G_j$ with $\\psi(g x) = f(g) \\psi(x)$ for all $g \\in G_i$ and $x \\in \\tilde U_i$.\nLet $x = \\varphi(\\tilde x)$ for a chart $(\\tilde U, G, \\varphi)$.\nUp to isomorphism, the stabilizer of $\\tilde x$ in $G$ does not depend on the chart; it is the *local group* of $x$.\nThe point $x$ is *singular* when its local group is not trivial.\nExample: let $m \\ge 2$ and let $G = \\Z / m\\Z$ act on $\\R^2$ by rotations through multiples of $2\\pi / m$.\nThe quotient $X = \\R^2 / G$ is Hausdorff and second countable.\nThe single chart $(\\R^2, G, \\varphi)$, with the quotient map $\\varphi$, is an atlas: clauses 2 and 3 only ask for the chart itself.\nSo $X$ is an orbifold, the cone with one cone point of order $m$.\nNon-example: let the group $\\Q$ act on $\\R$ by translations.\nEvery orbit is dense, so the quotient $\\R / \\Q$ carries the trivial topology and is not Hausdorff; it is not an orbifold.\nA [[smooth manifold|R2TQ82PCNB]] is an orbifold all of whose local groups are trivial.\n",
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